Is there a special case for equations of curves that have parallel oblique asymtotes?

Question

I am given the function $f(x,y)=(x-y)^2(x^2+y^2)-10(x-y)x^2+12y^2+2x+y$ and asked whether the curve $f(x,y)=0$ has two parallel oblique asymptotes, two unique asymptotes, or if there is no oblique asymptote at all. Do I manually find the asymptote equations or there is some shortcut to this?

Answer 1

The algebraic approach here is not too easy, so here's how to blend in some calculus (and some nontrivial computation, perhaps done with the help of a CAS). Of course, using a graphical CAS, you can easily see that there are two asymptotes with slope $1$. My original guess was that $y=x$ was an asymptote because of the leading factors of $x-y$. I was wrong.

Let $f(x,y) = (x-y)^2(x^2+y^2)-10(x-y)x^2+12y^2+2x+y$.

Using implicit differentiation, we can examine the limiting slope of the level curve $f(x,y)=0$ when $y=mx+b$ as $x\to\infty$. The leading behavior is $$\frac{dy}{dx} \approx \frac{m^3-2m^2+3m-2}{2m^3-3m^2+2m-1} = \frac{(m^2-m+2)(m-1)}{(2m^2-m+1)(m-1)},$$ and this has limit $m$ precisely when $m=1$. So we look for slant asymptotes $y=x+b$.

Let's review what it means for $y=x+b$ to be an asymptote of the curve $y=g(x)$. This should mean that the vertical error between the curve and the line approaches $0$, i.e., $$\lim_{x\to\infty} \big(g(x)-(x+b)\big) = 0.$$ Now, consider our level curve $f(x,y)=0$. Using linear approximation, we have $$0 = f(x,y) = f(x,x+b) + \frac{\partial f}{\partial y}(x,x+b)\big(y-(x+b)\big),$$ and so our vertical error is given by $$y-(x+b) = -\frac{f(x,x+b)}{\frac{\partial f}{\partial y}(x,x+b)}. \tag{$\star$}$$ Start plugging away, substituting $y=x+b$ (so $x-y=-b$): \begin{align*} f(x,x+b) &= b^2(x^2+(x+b)^2)+10bx^2+12(x+b)^2 + 2x+x+b \\ &= 2(b^2+5b+6)x^2+2(b^3+12b)x+\dots. \end{align*} This will be smallest (for large $x$) when the leading coefficient vanishes, i.e., when $b=-2$ or $b=-3$. In these cases, $f(x,x+b)$ will be given by a linear polynomial. On the other hand, considering ($\star$), $$\frac{\partial f}{\partial y}(x,x+b) = 2(2b+5)x^2 + \dots$$ is quadratic for both $b=-2$ and $b=-3$, so the quotient giving $y-(x+b)$ is the quotient of a linear polynomial by a quadratic polynomial, and hence goes to $0$ as $x\to\infty$.

Whew! In sum, the asymptotes are $y=x-2$ and $y=x-3$, as, indeed, @quasi commented.

Answer 2

An oblique asymptote occurs if $m:=\lim_{x\to\infty}\dfrac yx$ exists and is finite. Plugging $y=mx$ in the implicit equation and keeping only the quartic terms in $x$,

$$(1-m)^2(1+m^2)=0$$ has the double root $m=1$.

To check if the two asymptotes are distinct or not, we can plug $y=x+p$ and solve for $p$:

$$(-p)^2(x^2+(x+p)^2)-10(-p)x^2+12(x+p)^2+2x+x-p=0 $$

or, keeping only the quadratic terms in $x$,

$$2p^2+10p+12=0$$ which gives the roots $p=-2,p=-3$ and the oblique asymptotes

$$y=x-2,\\y=x-3.$$